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Theorem ofcval 28532
Description: Evaluate a function/constant operation at a point. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Hypotheses
Ref Expression
ofcfval.1  |-  ( ph  ->  F  Fn  A )
ofcfval.2  |-  ( ph  ->  A  e.  V )
ofcfval.3  |-  ( ph  ->  C  e.  W )
ofcval.6  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  B )
Assertion
Ref Expression
ofcval  |-  ( (
ph  /\  X  e.  A )  ->  (
( F𝑓/𝑐 R C ) `  X
)  =  ( B R C ) )

Proof of Theorem ofcval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ofcfval.1 . . . . 5  |-  ( ph  ->  F  Fn  A )
2 ofcfval.2 . . . . 5  |-  ( ph  ->  A  e.  V )
3 ofcfval.3 . . . . 5  |-  ( ph  ->  C  e.  W )
4 eqidd 2403 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
51, 2, 3, 4ofcfval 28531 . . . 4  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  A  |->  ( ( F `  x
) R C ) ) )
65adantr 463 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  A  |->  ( ( F `  x
) R C ) ) )
7 simpr 459 . . . . 5  |-  ( ( ( ph  /\  X  e.  A )  /\  x  =  X )  ->  x  =  X )
87fveq2d 5852 . . . 4  |-  ( ( ( ph  /\  X  e.  A )  /\  x  =  X )  ->  ( F `  x )  =  ( F `  X ) )
98oveq1d 6292 . . 3  |-  ( ( ( ph  /\  X  e.  A )  /\  x  =  X )  ->  (
( F `  x
) R C )  =  ( ( F `
 X ) R C ) )
10 simpr 459 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  X  e.  A )
11 ovex 6305 . . . 4  |-  ( ( F `  X ) R C )  e. 
_V
1211a1i 11 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  (
( F `  X
) R C )  e.  _V )
136, 9, 10, 12fvmptd 5937 . 2  |-  ( (
ph  /\  X  e.  A )  ->  (
( F𝑓/𝑐 R C ) `  X
)  =  ( ( F `  X ) R C ) )
14 ofcval.6 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  B )
1514oveq1d 6292 . 2  |-  ( (
ph  /\  X  e.  A )  ->  (
( F `  X
) R C )  =  ( B R C ) )
1613, 15eqtrd 2443 1  |-  ( (
ph  /\  X  e.  A )  ->  (
( F𝑓/𝑐 R C ) `  X
)  =  ( B R C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058    |-> cmpt 4452    Fn wfn 5563   ` cfv 5568  (class class class)co 6277  ∘𝑓/𝑐cofc 28528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-ofc 28529
This theorem is referenced by:  probfinmeasb  28860
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